Two computer romance games that I particularly like are mycandylove & starproject. (No, you don't have to download them or pay for them. They are free to play online.)
In my candy love you take on the role of a girl who is a new high school student. Throughout the episodes you have to help the other character's with their problems. Then at the end of the episode you get to go on a date with which ever guy you have the most candy love on your love o meter. The only problem is that it can take a lot of time and energy to finish an episode and you only get so many energy points each day you log on. So you can spend weeks on an episode.
In Star project you are a young woman whose parents once owned a star management company. However, they both died in a car accident and you must take over the company along with a lot of debt. Soon a young man appears at your company in hopes of becoming a star. You must work with him to help both your company and him to succeed. You can do fun things such as go on dates with him and such as well. The only down side to the game is you have a limited number of energy (it renews it's self at a good rate though, so you can play multiple times a day), and that it's in real time so if you schedule an appointment you need to be sure not to miss it. Other than that my only complaint is that later on you'll get a new star but he'll only work for you a while until you have to 'buy' him back (literally), because he needs a renewed contract.
There have also been some Ipod romance games that I'm interested in. Unfortunately, I don't have an Ipod, yet, so I can't really tell you how good they are. But, here are two romance games for the Ipod made by Voltage that look interesting. There are more than these.
Pirates in Love:
Be My Princess for GREE: