OK...Here's the deal, I am guessing this has only started happening since you replaced the t-stat. If so, there is a very good chance the problem is with wiring or the t-stat itself.
Over here in the great state of Utah (not sure about the rest of the country) when wire is ran for a t-stat, you have 4 wires from the furnace. Red, white, green, & blue. Now here is the tricky part, follow along closely:
The red wire is hot and gets hooked up to the red terminal on the t-stat. (not very hard right?)
The white wire gets hooked up to the white terminal on the t-stat. (still easy?)
The green wire controls your fan and gets hooked up to the green terminal on your t-stat. (that was easy)
Here is where it gets tricky, the blue wire is for your A/C. But it doesn't get hooked up to the blue terminal on the t-stat, it goes on the yellow terminal. (why not just use a yellow wire, right?)
If your t-stat is hooked up like that already, then check your jumper settings. Do you have a jumper going from the red terminal on the t-stat to another terminal (sometimes called rh & rc)? If so, try disconnecting that. Some systems don't need the jumper to make the system function properly.