A three-dimensional approach based on momentum vectors as variables for solving the three-nucleon Faddeev equation in first order is presented. The nucleon-deuteron breakup amplitude is evaluated in leading order in the nucleon-nucleon [Formula Presented] matrix, which is also generated directly in three dimensions avoiding a summation of partial wave contributions. A comparison of semiexclusive observables in the [Formula Presented] reaction calculated in this scheme with those generated by a traditional partial wave expansion shows perfect agreement at lower energies. At about [Formula Presented] nucleon laboratory energies deviations in the peak of the cross section appear, which may indicate that special care is required in a partial wave approach for energies at and higher than [Formula Presented]. The role of higher order rescattering processes beyond the leading order in the [Formula Presented] matrix is investigated with the result that at [Formula Presented] rescattering still provides important contributions to the cross section and certain spin observables. The influence of a relativistic treatment of the kinematics is investigated. It is found that relativistic effects become important at projectile energies higher than [Formula Presented].