There are several factors to be taken into consideration, such as skill level, inventory state, connection, gameplay style, map/faction and the most important: the team (if your team is good enough to make Platinum a DPS race, you rarely need Cyclonics anyway).
Let's assume things are overall pretty bad but one has unlocked at least the Reegar, Acolyte, Harrier and Black Widow. Some good no-Cyclonic characters in this case are the Juggernaut (Tank + Weapon Platform build); TGI, SI, AIU (little or no Fitness for these Infiltrators); and the Geth Trooper (Flamer build, you can consider running without Hunter Mode).
Drell and Vorcha make virtually no use of Cyclonics, so you can't really go wrong with them. However, they can be pretty squishy, the only kit among them I would recommend to a not so experienced player is the Drell Assassin.
There are several other characters that might be run without Cyclonics (rather a power amp or adrenaline mod) provided there is enough care and teamwork, such as: pretty much all Snipers, N7 Paladin, Batarian Soldier and Adept, Salarian Engineer, Turian Sentinel, Krogan Shaman, among others.