The parametrizations q = q(0) + q(1)z and q = q(0) + q(1)(1-a/a(0)) (Chevallier-Polarski-Linder parametrization) of the deceleration parameter, which are linear in cosmic redshift z and scale factor a, have been frequently utilized in the literature to study the kinematics of the Universe. In this paper, we follow a strategy that leads to these two well-known parametrizations of the deceleration parameter as well as an additional new parametrization, q = q(0) + q(1)(1-t/t(0)), which is linear in cosmic time t. We study the features of this linearly time-varying deceleration parameter in contrast with the other two linear parametrizations. We investigate in detail the kinematics of the Universe by confronting the three models with the latest observational data. We further study the dynamics of the Universe by considering the linearly time-varying deceleration parameter model in comparison with the standard Lambda CDM model. We also discuss the future of the Universe in the context of the models under consideration.