First off, you have to understand the first (and only) playing I've had so far was tainted in two ways. One, we badly mis-played a rule which made things waaaaayyyy out of whack. Second, Jerry played.
That aside for a minute, Senji is a combination of set collection, negotiation, and war. Basically, each player is a controlling house in feudal Japan. Each player gets a set of cards with different values (both the cards and the sets are different for each house). The cards are three types - hostages (your family), trade cards, and military support. Each player starts with three spaces on the board where you will place orders - as your empire expands or shrinks, you will place more or less orders each turn. Each player also starts out with three military units (all units are the same) and one unique Samurai. Each Samurai in the game has unique powers and a player can gain and lose them throughout the game. The player that has the Samurai with the highest honor starts the game "hosting the Emporer" - which means making all the decisions. Lastly, each player gets a starting hand of Hanafuda cards (the set collection cards used for trade). After that the player in the lead at the end of the round makes all the choices the next round.
Each round starts with 4 minutes (no more) of negotiation. This is not a lot of time in the least. Hostage cards are leverage to discourage another player from attacking you (or if you give the hostage, from you attacking them). They are leverage, because if you attack, the hostage can be executed which costs you points AND earns the other player points. Trade cards are guarantees that you will perform a trade with another player when called upon of your Hanafuda cards. Military support cards are powerful ally cards that buff the ally you grant them to. If you plan to attack, these cards will make a HUGE difference (and its the part we mis-played, which made the game crazy bad). You don't need military support to fight, but you don't want to go into battle without cards against someone else that has them. Given the rate at which we negotiated, 4 minutes flies by. Recognizing the military value now, I'd guess the 4 minute mark will seem even closer next time.
After the negotiation phase, each player secretly places orders in each area of the board they control. You have three options - get Hanafuda cards, get 2 armies in that area, or move (which generally means fight). You secretly place them ala A Game of Thrones - face down. Following the order placement, the guy in control (The Emperor's host) picks each province in whatever order (s)he wants and the player does the action. Obviously, 2/3 of the actions take about 5 seconds to resolve. Moving/fighting is actually pretty easy to resolve and involves a slight amount of luck (dice offer a 1/6 chance to give support to either) in addition to any military support that players want to use from other players. Win or lose, both armies are reduced by the same amount - the only difference being who gets the honor points (if the attacker wins, they get double).
The last phase of each turn is the trade phase. Here players will trade with each other, turn in sets of Hanafuda cards or whatever to score points, or recruit new Samurai. After the round is over, the new leader (if there is one) gains control and you rinse and repeat.
The beauty here is that there is actually a lot to do and pay attention to, even though things seem pretty simple. There are lots of ways to score points - Hanafuda cards, combat, and even simply getting other players to work deals with you (if you can acquire and not use 5 cards from other players - any 5 cards, you can return those cards and gain 10 points - 1/6 of what you need to win the game). I really hope this gets another shot at the table. I'm not sure that everyone will care for it - it really is a negotiation game and one that could potentially bog down a little if folks aren't doing much each turn. However, if you aren't trading cards with the other players, you probably aren't giving yourself a chance to win.