Basically you have two options depending on how much work you want to do.
1) You can upgrade to version 5.7. This is the latest 16 bit version of dBASE. Depending on the complexity of your application, you *may* be able to run it from within a Dos emulation window. There will be some concerns on this however, since dBASE III had very different formats for screens and reports. These things will probably have to be rebuilt.
dBASE 5.7 comes with a utility called a Component Builder. The Component Builder is designed to take old dBASE IV for Dos applications and rebuild them in the 5.7 environment. Again, depending on the complexity of your application, it may be able to convert your dBASE III applications.
2) You can upgrade to version 7.5. This is the latest 32 bit version of dBASE. Upgrading to this version will be more difficult however. Your database tables will be readable, but virtually everything else would have to be recreated. That is a lot of work, but the advantage to this is that your applications can be built fully object oriented. This means greater stability for your applications in the long run.
Ultimately you're going to have to do a fair amount of work retooling your existing applications. dBASE III was created such a long time ago that there is not much in the way of support for it anymore. One other thing I can recommend however, is that you look into our newsgroups on the web. We do have a group specifically devoted to older versions of dBASE and the people who are there can answer your questions as well as help provide support for your actual migration. Talk with them. Many have already done what you are looking into doing and they can help you figure things out.
Information about the dBASE, Inc. *free* newsgroups can be obtained at: