I don't think you'll find too many "hard core" RTTY ops using outboard multi-mode controllers any more. Without a doubt, the most popular RTTY sound card program is MMTTY. While the RTTY part of DM780 is actually pretty good for everyday rag chewing, its major weakness is that it will only do RTTY in AFSK (Audio Frequency Shift Keying) mode. (I think HRD v6 will do FSK RTTY however). That is probably the biggest advantage of MMTTY is that it will do FSK which is what most serious RTTY ops prefer.
And while the whole HRD suite is excellent (I use it!), a contester will use a dedicated contest logging program. One of the most popular is N1MM. Not only is it very robust, but it also happens to be free. Why use a dedicated contest logger? Contests run for a limited amount of time and high scoring contesters need to make the best use of that time and one way of doing that is by not wasting time contacting a station he has already contacted and the program will display the callsign of a station you have worked and stations you have not worked in different colors so you can instantly tell whether or not you should try and contact that station.
Also, a good contest logging program will have a template for each contest so that you are only logging the information necessary for that contest. For contests such as the RTTY Roundup, all you need is a signal report (which will ALWAYS be 599 in a contest!) and the other station's state and that is it.
And this may seem like a, "Well duh!" comment, at the end of the day, the radio itself is what is going to do the "heavy lifting". While any radio can do AFSK RTTY, not every radio can do FSK, especially entry-level radios. Radios that run FSK will generally have a separate set of RTTY filters specifically for RTTY that can make a world of difference. Icom, for one, does a great job of including features for RTTY on some of their rigs. I am sure other brands do quite well, but I happen to own an IC-7410!
If terms like AFSK and FSK confuse you, as others have suggested, you should check out AA5AU's website. http://www.aa5au.com/rtty.html
He is without a doubt one of the most knowledgeable RTTY contesters in the world so you can take his word that he knows what he is talking about.
See you in the next contest!