While not as familiar with the routes east of Ohio, here are a few variations as routed through Google. The first is the recommended route for timing and staying on major (4-lane) roads for the majority of your trip. It shows 2,902 miles from Burlington to L.A.
The challenges with that route in mid-March are weather, especially through Colorado and along the Great Lakes. Iowa and Nebraska are fairly boring for the most part, as is much of the "bread basket" (Kansas, parts of Oklahoma, etc.). Like others have said, Chicago is a great city but you will have traffic and possibly weather to contend with. If you are more concerned with getting there as quickly and safely as possible then go further south, like this:
While it only adds 35 miles (showing 2,937), it avoids some of the more heavy traffic around Chicago, and should give you better weather as you start heading south. Albuquerque to L.A. has some neat scenery and you can zip over to the Grand Canyon and/or Las Vegas without a huge hassle. It has regular stops with "branded" hotels (Hilton, Marriott, Holiday Inn, etc.) along the way. Then there is the route you suggested, only bypassing Chicago and going more directly:
This one shows 2,968 miles, including a whopping 1,713 miles on I-70 alone. Again, Kansas is boring with 70 MPH speed limits heavily enforced, while Colorado can be tricky depending on weather. Mountain roads, even the well-maintained interstates, can be a serious challenge in bad weather. While daytime temperatures in March can get warm enough to keep the roads clear, the freezing temperatures at night can turn snow melt into ice - even with heavily salted roads. If they are getting heavy snows, they can enact the "chain law" whereby you must have chains on your vehicle to drive through certain passes - even on major interstates like I-70.
If the weather looks clear and has for a week beforehand (allowing snow to melt on the roads, etc.) it is a breathtaking drive. You are still likely to encounter salt/sand on the roads, so don't plan on any triple-digit speeds on mountain corners anytime soon. Also note that once out of Grand Junction, you have a long stretch of "no-mans" land past Green River Utah - over 100 miles with no services and spotty cell service. Pretty country to be sure, but not where you want to get caught with a flat or even needing a bathroom with running water. Once on I-15 in Utah, you will find some of the few 80 MPH speed limits in the country. It can still have fairly heavy traffic with folks heading to Vegas, especially towards the weekends. If you are getting to the Vegas area during the day, do a side trip through Valley of Fire State Park off I-15 starting at Moapa Valley on Highway 169. It might add a couple hours, but is a beautiful winding drive that loops back to I-15.
For scenery you will be hard-pressed to beat going through the Rockies in Colorado and some of the other ranges in Utah. If weather is a concern and you haven't seen the Grand Canyon, the more southern route is also very good with some interesting side attractions - Petrified Forest National Park is also interesting and that road can be fun in a sports car.
Make sure your car is up to ***** - fresh fluids, good tires, and an emergency tire kit like Fix-A-Flat or something assuming you aren't on original runflats. Bring maps and/or GPS, binoculars, and a good digital camera with a quick power-up to that first shot: Long trips can provide some exciting photo opportunities, especially with some of the bird migrations during this time of year. Look for bald eagles along many of the river valleys, as well as major flocks of many other species. Watch for animals on the roads, including elk on all the western routes (north or south) and mountain goats in Colorado - especially at night, as they tend to get closer to the roads (and salt) when they are harder to be seen.
Have a GREAT time and be sure to let some of the regional clubs know if/when you are passing through - you just might catch a Viper gathering or two along the way!
)
They built the trunk too small. 
