Find Las Ramblas (famous/touristy street in downtown) and run down it in the morning towards the harbor. Make a left (south) and you can run on the various bike paths, etc. along the harbor and beaches, to the convention centers/old Olympic rowing venues and beyond. Good for long runs/easy runs.
Other long run option (will take awhile to get there): La Carretera de las Aguas is a long/measured dirt road that runs the length of the hills behind Barcelona. Look for it on a local map or google earth. Great run, totally flat, long, and a favorite for local runners and cyclists. Also many offshoot trails and a great place for mtn biking. Getting there can be tricky; the best bet is to take the fenicular up that leads to the amusement park on the top of the hill, but stops halfway (where the carretera is).
Interval workouts: find Montjuic (where the olympic stadium is). There is a track in the same general vacinity where you can pay to run on (and grab a capuccino after at their cafe). There are also several running clubs that meet for intervals there of all talent levels where you may be able to jump in a workout or two. If you're driving (or there is another fenicular) or don't mind hills, you can also jog up to the castle on top of Montjuic and run intervals within the walls of a 15th structure. Totally open and not alot of traffic even in the afternoons up there. I had a few 1k and 2k loops up there I did for interval training. You'll also see a track if you run along the harbor; all tracks require payment, but you can run on this one as well during its operating hours.
There is also a park near downtown--don't remember the name of it, but there are a series of dirt paths and grass that several running clubs in town use for intervals.
Depending on the hours you keep, I'd also recommend an early morning jog through the gothic quarter off of Las Ramblas. Fun to trot through gothic corridors without the conjestion you'll experience during the day.
These runs will keep you away from alot of the car traffic (except getting to/from these locations). I'd recommend going early morning, as heavy traffic begins around 6-7 and dosn't stop until 3 the next morning. If you're a male, I wouldn't hesitate to run in any of these locations alone. If you're a female, I'd have some company--especially if running in Montjuic or up on the carretera at any time of day. If you're taking subways, wear sweats or longer clothing over running shorts/tank tops unless you're cool with gawking/stares; this goes for male/female both. If/when you stash said sweats to go for a run, stash them good. Barcelonians have a tendancy to think that anything left--even in a hidden location--is there's to take. Depending on how long you're staying and your tranportation, there are a ton of out-of-the-way places to run on country roads and trails where you really get a feel for Catalonia, and only a few miles from the city limits. Many runs can start from around the amusement park up on the hill (forget its name) and essentially lead you on lightly traveled dirt roads to the next town/village; you'll think your're in the 18th century sometimes. Definitely a bit more involved to scout good runs out using local maps and google earth, but well worth it if you have the time and like those exploratory type of runs.
Lived there for a year, so let me know if you have any other questions, and have a great time. Great city for visiting!