If you throw "LED MR16" at Google, you'll see lots of solutions on offer, but many are poorly made. LED vendor Cree has built a LED module expressly for this market. Hunting around, there's at least one vendor claiming to have this in production, but I can't find anywhere to actually buy it.

This is a rapidly evolving market. My intent is to wait a year or two for these things to shake out, then go for it.

Warning: if you do choose to buy one of these things now, pay careful attention to two numbers: the color temperature (reported in degrees Kelvin) and the CRI (color quality, with 100 being "perfect"). If you want to match your existing halogen bulbs, then you want a color temperature of 2700K and as good a CRI as you can find (80 minimum).

EDIT: Warning #2: one other thing worth waiting for is that newer controller boards for these LED MR16 fixtures will be compatible with the electronic dimmers that are commonly used.