The multiple knapsack problem with grouped items aims to maximize rewards by assigning groups of items among multiple knapsacks, considering knapsack capacities. Either all items in a group are assigned or none at all. We propose algorithms which guarantee that rewards are not less than the optimal solution, with a bound on exceeded knapsack capacities. To obtain capacity-feasible solutions, we propose a binary-search heuristic combined with these algorithms. We test the performance of the algorithms and heuristics in an extensive set of experiments on randomly generated instances and show they are efficient and effective, i.e., they run reasonably fast and generate good quality solutions.