We've got some Honeywell CT1950 electric heat thermostats. According to the instructions, the black wires go to the line voltage and the red wires go to the heater.

Probably not. The incoming power goes to L1 & L2. That will be a black and a red most likely.
Please describe the wires. Is there two cables, each with a black and red wire? If so then one of them is the incoming power which goes to L terminals. The other one is the heater connection which goes to the T terminals.

I'm thinking along joed's lines, myself.

In the first place, relying on wire color alone is not altogether safe because it depends only on what kind of wires were used by whoever wired the original.

Most likely, the original thermostats are two-pole, 240V line voltage switches. I'm guessing that because there are two black and two red wires (no whites--which might tend to suggest neutrals present, and possibly, a single-pole 120V switch).

But rather than "assume" (and we all know what that stands for), it'd be better to understand the circuitry. One method is to trace the wires and determine which two are line supply wires (i.e., provide power from the panel) and which two are load feeds (i.e., go directly to the heater elements). Another way is to use a meter or tester to verify incoming 240V supply.

Keep in mind that some thermostats only switch one pole. That's true, for example, with a Honeywell CT1950A--it is a single-pole switch. The Honeywell CT1950B is a two-pole switch. (See this PDF for details.)

Although those particular instructions do show the lines as two blacks and the loads as two reds, that's somewhat puzzling. I agree with joed's point that the two line connectors are at least as likely, if not more likely, to be red and black and the two loads also red and black. (In still other cases, one or more of the cables may be black and white, even if it is a 240v circuit!)

Bottom line: let's make sure we understand the circuitry before making a recommendation.

One other point in passing: Electric heaters can pull a quite a bit of power and it's important to pay attention to the amperage ratings of the thermostat/switches. Can you verify that your replacement thermostats have (1) enough capacity to serve the total demands of the heaters they serve and/or (2) that they are as large or larger than the thermostats they are replacing?

Well, I was actually able to find a diagram on the back of one of the face plates on the old thermostats. Looks like the originals were wired red to line, black to load. Now the back of the new thermostats has one red and one black wire on each side labeled L1 T1, and L2 T2.

Here is the thermostat that I bought:
http://yourhome.honeywell.com/Consumer/Cultures/en-US/Products/Thermostats/Non-Programmable/Non-Digital/Do-It-Yourself/LineVoltage/Default.htm

Hope this link works. The model # is CT62b by Honeywell.

I'm not all that familiar with electrical wiring, so thanks for all your help.