The time dependencies of the carrier relaxation in modulation-doped InGaAs-GaAs low-dimensional structures with quantum wires have been studied as functions of temperature and light excitation levels. The photoconductivity (PC) relaxation follows a stretched exponent with decay constant, which depends on the morphology of InGaAs epitaxial layers, presence of deep traps, and energy disorder due to inhomogeneous distribution of size and composition. A hopping model, where electron tunnels between bands of localized states, gives appropriate interpretation for temperature-independent PC decay across the temperature range 150-290 K. At low temperatures (T < 150 K), multiple trapping-retrapping via 1D states of InGaAs quantum wires (QWRs), sub-bands of two-dimensional electron gas of modulation-doped n-GaAs spacers, as well as defect states in the GaAs environment are the dominant relaxation mechanism. The PC and photoluminescence transients for samples with different morphologies of the InGaAs nanostructures are compared. The relaxation rates are found to be largely dependent on energy disorder due to inhomogeneous distribution of strain, nanostructure size and composition, and piezoelectric fields in and around nanostructures, which have a strong impact on efficiency of carrier exchange between bands of the InGaAs QWRs, GaAs spacers, or wetting layers; presence of local electric fields; and deep traps.