Here is my situation:
Last winter we started having a problem, then it got warm, we switched to A/C and no problems. Now its cold again and our problem starts again. We have a digital thermostat. We set the desired temparature to 69 for example, the furnace kicks on, warms everything up, and of course, shuts off like it should. The problem starts when the temp drops below our desired setting. The furnace does not kick back on again. I had it get all the way down to 61 recently. We have to turn the desired temp down to 1 degree below whatever temp the thermostat says it currently is, then it kicks on. Then immediately raise the desired temp from whatever that was, say 60, up to what we want, usually 69. It warms back up to above 69 and the whole process starts again.
I have someone wanting to replace the thermostat, but I wasn't sure that was the main problem since we had no problems with it in the summer. I would think that if the therm was the problem we would have maybe had the reverse in the summer and the house would have gotten too hot without the A/C kicking on. But... it worked fine then.
I saw a similar question but the people there said they had to reset their therm to make it work again and also the furnace shut off at other times when it shouldn't. Ours doesn't do that.
Any suggestions? Thanks in advance!

I forgot my crystal ball today and I will have to depend on you to post the EXACT model number and brand of thermostat. Since over the years they have made 5000 different ones It is tuff to guess the correct one you have without my crystal ball..

Actually after describing my whole deal I somehow forgot most of that important stuff... I was hoping I beat anyone with an answer like yours back here... :) guess not
I am actually at work getting ready to go home but it is a gas furnace in case anyone else wants to know and ill have to get the right brand for the furnace at home. It's only 3 yrs old though came w/ the house when it was built. The therm is a RiteTemp Model GPMG8035C, seen here RiteTemp 7 Day Programmable Off White Large Vertical Touch Screen Thermostat Energy Star Compliant - GPMG8035C at The Home Depot (http://www.homedepot.com/webapp/wcs/stores/servlet/ProductDisplay?id=100045786&jspStoreDir=hdus&catalogId=10053&marketID=401&productId=100045786&locStoreNum=8125&langId=-1&linktype=product&storeId=10051&ddkey=THDSiteMap)

Install guide.

http://www.ritetemp-thermostats.com/images/8035C_installation_guide.pdf

Operation guide

http://www.ritetemp-thermostats.com/images/8035C_operation_guide.pdf

Read both to see if your solution is there. Just in a short glance there is info on swing in one manual and system type settings in the other.

Before anyone asks we went through the entire troubleshooting guide for the specific thermostat, including the reset button on the thermostat. The differential is set for 2 degrees. We had it drop all the way down from 70 to 61 before without kicking back on.
What gets me is that all it takes to start the furnace back up is setting the desired temp 1 degree below the current room temp and then turning it back up again. I just can't wake up every 30 minutes at night to do it. :)

You must have typed that right as I was typing again... I'm too slow

Also confusing to me is that with the A/C in the summer there were no problems

In that case you probably have a bad thermostat since you already tried to adjust.

Here is exactly what happens:
A couple nights ago we went to sleep w/ the thermostat set at 70 degrees, it had been off all day since it was fairly warm out. Before going to sleep the temp in the house was 67, we turned it on heat from off and set the temp to 70, the furnace kicked on and stayed on until the room warmed to the right temp and we went to sleep. We woke up I looked at the thermostat and the temp in the house was 61. I hit the down arrows to get the desired temp down to 60 then immediately hit the up arrows - as soon as I got to 62 the furnace kicked on... I kept going up all the way to 70, and by the time I left for work it was 70 degrees in the house. When I got home it apparently had not come back on all day and it was cold again.

I would have just thought that we would have had the same problem with the A/C in the summer if the thermostat itself was bad. On a whim I turned the air on yesterday and everything works perfectly. It was cold but it worked... haha.
I turned the crazy thing to air, set it at 66 while it was 69 in the house, and the air kicked on. It kicked off when it got really horribly cold, the room got warmer - because I had a space heater going - the air kicked back on...
Could the thermostat decide that it only likes the A/C and not the heat?

Next time it happens, just gently bump the thermostat instead of trying to change the setpoint. See what happens.

Will try as soon as I get home and post back

Yes it is possible for a t-stat wo work on A/C correctly and not heat. If you have gone and done all that is in the manual and it still does not work then it must be bad. Unless somehow it is going back to a preset program the factory or you set.