You do not need to be a total asshole to everyone. In fact, you can do this with very high Paragon and very low Renegade. You just have to make sure you respond negatively at certain critical points.
Nearly all characters who were squadmates in ME2 can only be romanced in ME3 if you carry over a romance from the previous game. Most of them will, at some point, ask Shepard how they feel about the relationship and whether to continue it, and this is by far the easiest point to turn them down. If you're feeling cruel, you can often string them along for a while and then turn them down at a later point of no return.
It should also be noted that all romanceable characters have a romantic point of no return. Once you've gotten one character past that point, they are "locked-in," and all other romances will automatically terminate. This isn't very helpful if you're trying to get through the game with no romances at all, of course, but it's still important to understand how the game is played. This is a simplification from the previous games, in which characters could become jealous of one another if you tried to romance more than one at a time.
Of the remaining characters, male Shepard has two exclusives:
- Ashley: She is most easily avoided by not taking her aboard the Normandy, but that is not required. The wiki tells me there is a cutscene on the Citadel where she asks whether Shepard is serious about her; I imagine this is largely identical to the Kaidan romance below.
- Steve: When you meet him at Purgatory, choose the option that suggests Shepard is straight. Unfortunately, Bioware does not (to the best of my knowledge) allow you to avoid romancing Steve while also roleplaying a Shepard who isn't straight. This is particularly stupid given that you can immediately turn around and romance Kaidan (if he's alive etc.).
Female Shepard has three exclusives:
- Traynor: When you invite her up to your cabin and she asks to play chess, reply "That's it?"; this looks like it might initiate a romance, but it doesn't.
- Vega (Citadel only): At one point during Citadel, Shepard will begin flirting with him a bit more aggressively than their usual banter. He will be surprised and ask if she means it. Say no.
- Javik (Citadel only): To the best of my knowledge, this is unavoidable, but it is a single cutscene and does not significantly affect the story.
Both Shepards have these options:
- Kaidan: Like Ashley, you can refuse to take him aboard, but you don't have to. There is a cutscene on the Presidium where he and Shepard get lunch; if you are not locked in with anyone, he will ask how Shepard feels about him. This is the point of no return for his romance.
- Liara: She may ask to continue a romance from a previous game shortly after the first Priority: The Citadel or so, and can be neatly declined at this point. But the point of no return is the cutscene on the Presidium where she talks about Benezia.
- Allers: When she very transparently tries to flirt with you after the first interview, turn her down and she'll pretend you misinterpreted her and never bring it up again. Alternatively, don't let her onto the Normandy in the first place.