A multicarrier-division duplex (MDD)-based cell-free (CF) scheme, namely MDD-CF, is proposed, which enables downlink (DL) data and uplink (UL) data or pilots to be concurrently transmitted on mutually orthogonal subcarriers in distributed CF massive MIMO (mMIMO) systems. To demonstrate the advantages of MDD-CF, we firstly study the spectral-efficiency (SE) performance in terms of one coherence interval (CT) associated with access point (AP)-selection, power- and subcarrier-allocation. Since the formulated SE optimization is a mixed-integer non-convex problem that is NP-hard to solve, we leverage the inherent association between involved variables to transform it into a continuous-integer convex-concave problem. Then, a quadratic transform (QT)-assisted iterative algorithm is proposed to achieve SE maximization. Next, we extend our study to the case of one radio frame consisting of several CT intervals. In this regard, a novel two-phase CT interval (TPCT) scheme is designed to not only improve the SE in radio frame but also provide consistent data transmissions over fast time-varying channels. Correspondingly, to facilitate the optimization, we propose a two-step iterative algorithm by building the connections between two phases in TPCT through an iteration factor. Simulation results show that, MDD-CF can significantly outperform in-band full duplex (IBFD)-CF due to the efficient interference management. Furthermore, compared with time-division duplex (TDD)-CF, MDD-CF is more robust to high-mobility scenarios and achieves better SE performance.