We develop an adversarial-reinforcement learning scheme for microswimmers in statistically homogeneous and isotropic turbulent fluid flows, in both two (2D) and three dimensions (3D). We show that this scheme allows microswimmers to find non-trivial paths, which enable them to reach a target on average in less time than a na\"ive microswimmer, which tries, at any instant of time and at a given position in space, to swim in the direction of the target. We use pseudospectral direct numerical simulations (DNSs) of the 2D and 3D (incompressible) Navier-Stokes equations to obtain the turbulent flows. We then introduce passive microswimmers that try to swim along a given direction in these flows; the microswimmwers do not affect the flow, but they are advected by it. Two, non-dimensional, control parameters play important roles in our learning scheme: (a) the ratio $\tilde{V}_s$ of the microswimmer's bare velocity $V_s$ and the root-mean-square (rms) velocity $u_{rms}$ of the turbulent fluid; and (b) the product $\tilde{B}$ of the microswimmer-response time $B$ and the rms vorticity $\omega_{rms}$ of the fluid. We show that, in a substantial part of the $\tilde{V}_s-\tilde{B}$ plane, the average time required for the microswimmers to reach the target, by using our adversarial-learning scheme, eventually reduces below the average time taken by microswimmers that follow the na\"ive strategy.