A participant in an Alternate History mailing list asks the question:

Perhaps by this [medieval] time the Muslims would also be struck with the idea of sailing westward across the Atlantic.  From a religious point of view this could mean more land to make converts in.

Muslims during the mid-Middle Ages were as acquainted with ancient Greek philosophers as the Europeans were later to become.  Aristotle pointed the way west, for any willing to hear it, at the end of his proof of the sphericity of the Earth (still as valid today as when written in the 4th century BC).  It's worthwhile reviewing Aristotle's words, which incidently demolish the prevalent myth nowadays that cultivated ancients believed the world was flat.  After mentioning several logical arguments for the Earth's sphericity (which have not held up over time), Aristotle wrote: ¹

The evidence of the senses further corroborates this.  How else would eclipses of the moon show segments shaped as we see them?  As it is, the shapes which the moon itself each month shows are of every kind — straight, gibbous, and concave — but in eclipses the outline is always curved: and, since it is the interposition of the earth that makes the eclipse, the form of this line will be caused by the form of the earth's surface, which is therefore spherical.  Again, our observations of the stars make it evident, not only that the earth is circular, but that it is a circle of no great size.  For quite a small change of position to south or north causes a manifest alteration of the horizon.  There is much change, I mean, in the stars which are overhead, and the stars seen are different, as one moves northward or southward.  Indeed there are some stars seen in Egypt and in the neighborhood of Cyprus which are not seen in the northerly regions; and stars, which in the north are never beyond range of observation, in those regions rise and set.

All of which goes to show not only that the earth is circular in shape, but also that it is a sphere of no great size: for otherwise the effect of so slight a change of place would not be so quickly apparent.  Hence one should not be too sure of the incredibility of the view of those who conceive that there is continuity between the parts about the pillars of Hercules and the parts about India, and that in this way the ocean is one.

Beyond that initial beckoning of the way, I've run into a couple of references to actual expeditions west across the Atlantic prior to Columbus (and I don't mean the Vikings!), originating from Iberian or Mediterranean ports.  One such mention is Samuel Eliot Morison's The European Discovery of America, where in discussing the lead-up to Columbus's voyage Morison wrote: ²

When [Columbus] had learned enough Latin to read ancient and medieval cosmographers, he ascertained that Aristotle was reported to have written that you could cross the Ocean from Spain to the Indies paucis diebus, in comparatively few days; and Strabo recorded that certain Greeks or Romans had even tried it but returned empty-handed “through want of resolution and scarcity of provisions.”

The second mention of earlier voyages I've seen is Timothy Ferris's Coming of Age in the Milky Way, in which Ferris wrote: ³

Columbus's plan appeared foolhardy to anyone who possessed a realistic sense of the dimensions of the earth.  To sail westward to Asia, as the geographers of the court at Castile took pains to inform Columbus, would require a voyage lasting approximately three years, by which time he and his men would surely be dead from starvation or scurvy.  The voyage had been attempted twice before, by Moorish explorers out of Lisbon and by the Vivaldi brothers of Genoa in the thirteenth century; none had been heard from since.

I'm trying to gather more information about any pre-Columbian expeditions west.  If anyone has additional information, or is aware of a halfway serious study concerning them, please let me know!
 

References

¹ Aristotle, "On the Heavens," Book II, Chapter 14, The Works of Aristotle, Oxford University Press; pp. 297-298.

² Samuel Eliot Morison, The European Discovery of America: The Southern Voyages AD 1492-1616, Oxford University Press, New York, 1974; p. 17.

³ Timothy Ferris, Coming of Age in the Milky Way, Anchor Books/Doubleday, Bantam Doubleday Dell Publishing Group, Inc., New York, 1988; p. 56.
 

There will come a time in the later years when Ocean shall loosen the bonds by which we have been confined, when an immense land shall be revealed... and Thule will no longer be the most remote of countries.

Seneca

Everyone using Blogger is going through pains as the transition to “New Blogger” proceeds, replacing Blogger's old problems (which one could usually work around) with an entirely new set.  As I put it recently in an e-mail to Donald Sensing:

I'm not that happy with New Blogger, personally, so far.  The old Blogger never ate my archives (of which I have a local copy anyway), and I was always able to work around its problems.  The New Blogger issue of apparently only allowing a maximum of a couple of screen-fulls of text in each posting is potentially a killer for me though; hopefully they'll fix it soon.

Sensing has since run into another problem with New Blogger, whereby “Blogger eats posts.”  Bill Hobbs replied to Donald's concerns in One Hand Clapping's Comments section with the following words of advice:

The new improved blogger sometimes isn't.  I always highlight and copy my text before I hit the post button, just in case.

And for longer stuff, I build it in MS Word, or I enter a bit and post then add some and post... and add some and post... and think about moving to Movable Tybe... and add some and post...

Bill's advice is so sensible, in fact, that I thought I'd add a more extended commentary to flesh it out a little.  Sensing, of course, is far from a newbie (and has since noted that the lost posting was only a few lines long, and he normally does use an off-Blogger editor so postings can be separately saved).  However, the issues are common, and experience gained through hard knocks is usually painful.  I believe Bill's advisory applies well beyond just the world of Blogger, or blogging, into how to act effectively in a “noisy” (read: real world) environment.  Following is my commentary from One Hand Clapping:

I'd like to emphasize what Bill Hobbs said above.  It's a cliche that newbies (I don't mean you Donald!) fail to save while writing an entire document, or don't backup their hard drive, and then weep tears of frustration when the system or hard disk crashes.  We all know (from bitter experience) that's STOOPID — you have to save and backup your work.  Why do folks think one can do otherwise with blogs, or that the server can be trusted to save it perfectly (even if New Blogger is successful)?

I second Bill on using MS Word while writing a posting.  Word is convenient for building postings because (in addition to locally saving the document) Word's outline mode is handy while writing it, and macros are available to manipulate it line by line (or whatever), which has been useful for me from time to time.

After typing a posting into Word, it takes only a handful keystrokes to save and post it — Ctrl-S to save it; Ctrl-Home, Shift-Ctrl-End to highlight the whole thing, Ctrl-C it into the clipboard, Alt-Tab over to Blogger (or whatever the interface), and Ctrl-V to paste it in.

Bill's other point — on sometimes making only a few trivial little changes, saving, posting, looking at the results, repeat — is also very important.  The iterative procedure may not be needed if you're just building a text posting, but if one is doing anything fancy with HTML, or making more than a small change to your template, say, it's all too easy to break things drastically so that, sometimes, nothing shows up.  If you've made numerous changes (and discarded the previous version of the document or template to boot), the task of identifying and undoing the error can be extremely frustrating and unpleasant.

The elementary solution is to hold onto the previous version and make only a few alterations before trying out the new one.  If things go haywire, reintroduce the changes one by one until it fails, then you know where it's going wrong and what needs to be fixed.  Or, you can simply revert back to the earlier version and forget it!

Charles Austin replied on the thread, noting “One must be careful about the use of fonts in MS Word when cutting and pasting into Blogger.  Not all characters in all fonts are recognized.”  Charles' point is well taken, and my advice would be to disable some Word features, under “AutoFormat as you type” (pull down “Tools,” then select “AutoCorrect”), such as “smart quotes” and symbol characters.