We present the detailed growth and characterization of novel GaSb-based semiconductor saturable absorber mirrors (SESAMs) operating in the 2–2.4 µm spectral range. These SESAMs at different wavelengths are bandgap engineered using ternary material compositions and without strain compensation. We observe that even when the thickness of quantum wells (QWs) exceeds the critical thickness we obtain strain relaxed SESAMs that do not substantially increase nonsaturable losses. SESAMs have been fabricated using molecular beam epitaxy with a AlAs_0.08Sb_0.92/GaSb distributed Bragg reflector (DBR) and strained type-I In_xGa_1-xSb or type-II W-like AlSb/InAs/GaSb QWs in the absorber region. All the type-I SESAMs show excellent performance, which is suitable for modelocking of diode-pumped semiconductor, ion-doped solid-state, and thin-disk lasers. The recovery time of the type-II SESAM is too long which can be interesting for laser applications. The dependence of the SESAM design, based on its QW number, barrier material, and operation wavelength are investigated. A detailed characterization is conducted to draw conclusions from macroscopic nonlinear and transient absorption properties at different wavelengths in the 2–2.4 µm range for the corresponding devices.