1. Draw a diagram.
LSAT prep companies often say you need to draw a grid for each game, rather than a simple slot diagram. What is a slot diagram? I describe it in detail in my free subscribers-only Logic Games tip sheet. I also demonstrate it at my regularly-held free LSAT workshops. However, I'll briefly explain: it's a simple way to map the information in linear and combination games (games with both a linear and grouping element). Grids are time-consuming, but drawing slots for each letter (in a six person/thing game, it looks like: _ _ _ _ _ _ ) requires less space and time.
2. Use diagrams from earlier questions.
Keep your diagrams from the first few questions of each game because they will often save you time in later questions. If you haven't noticed this yet, try something new. Draw a separate diagram for each "if" question next to the answer choices for that question. Use the space at the bottom of the page to sketch your main diagram, which should contain your inferences from all the rules. In the vast majority of games, at least one previously-drawn diagram will help you solve a later question in a particular game. This lets you get through the question without having to draw a brand-new scenario.
3. For all "If" questions, draw a diagram before you look at any answer choices.
If you've look at any LSAT logic game, you'll probably see a question like the following: "If R is placed in the third position, which one of the following must be true?" or "If S is placed last, it could be true that..." I tell my students to stop reading the question right after the first half of the sentence (where the comma is) and to immediately sketch what must be true. More often than not, this bare-bones sketch alone will lead you to the correct answer. Rather than reading through the answer choices and trying each one out, you have effectively predicted the correct answer, saving valuable time.
4. Apply each rule, one-by-one, to the answer choices in the general "acceptability" questions.
The majority of games start with a question like "Which one of the following is an acceptable ordering / grouping / assignment..." etc. Four of the choices will each contain a scenario violating one rule or another, while only one will be acceptable. There are two ways to attack these questions. The first is the slow way - you might look at choice "A" and see if it follows each of the approximately five rules of the game. If it doesn't, then move to choice "B", etc. However, this approach requires you to go back and forth between the rules and the choices, costing you over a minute for what is typically a game's easiest question. The more efficient method is to take an "assembly-line" approach to this question type. Take the game's first rule and check it against each answer choice. After (hopefully) eliminating one or two choices, take the next rule and apply it to the remaining answer choices.
5. When the possibilities in a game are limited, map them all out.
Sometimes a game's rules interact in very restricting ways. For example, in Game 2 of PrepTest 37 (June 2002), which is about placing seven trucks in a certain order, the possibilities for the first three spaces become severely restricted after placing the other four trucks. Trucks X, Z, and U are the only ones left to be placed. Because Z must come before U, the only options are "X Z U", "Z X U" or "Z U X". Of course, you could just map one possibility, and if that doesn't work, start another. However, many students end up trying one possibility, and then they become frustrated when it doesn't work, wasting valuable time.
It's faster to sketch out the "skeleton" or "must be true" aspects of each possibility and then see which of those might actually work. I recommend listing each possibility immediately and then going through the game's questions. The bottom line: do one type of task at a time (diagramming each possibility, then reading the answer choices for each question). This will significantly decrease the amount of time you spend on each game.