Mode-locking of a directly diode-pumped Tm:LuAG laser is demonstrated using GaSb-based semiconductor saturable absorber mirrors (SESAMs). Stable and self-starting mode-locked operation was realized, generating pulses as short as 13.6 ps at 2024 nm with a maximum output power of 98 mW. Two GaInAs-based SESAMs were used for comparison with the operation based upon the use of the GaSb SESAM; in this case, longer pulses with durations of 27 ps and 34 ps were obtained under the same experimental conditions. Our work sets a new record in pulse duration for mode-locked Tm:LuAG lasers and confirms that lattice-matched GaSb-based SESAMs are beneficial for mode-locked solid-state lasers in the 2 μm range.