I chose a Whirlpool microwave when we built this house 12 years ago. Besides showing some finger smudge darkening on the keypad, it still works great.
Most brands are manufactured by just a few major manufacturers anymore, and then they're branded and sold under different names. But, just because the appliance says "Kenmore" it doesn't mean that it was made by LG. Electrolux/Fridgidaire also manufactures products branded with the Kenmore name.
The GE and Kenmore microwaves were most likely made by two different manufacturers (GE and LG respectively), but to make sure you'd need to check the appliance's model number with the list posted on the Appliance 411 website.
If you purchased the appliance from Sears, refer to this page to find your serial number prefix and it'll point out the manufacturer:
http://www.appliance411.com/purchase/sears.shtml
This is a link to the general appliance "Who Made What?" list:
http://www.appliance411.com/purchase/make.shtml
Hope this helps you find a replacement that won't give you trouble in the future!
*Tink