Field and laboratory observations of seismic wave propagation and attenuation are usually explained using the viscoelastic (VE) model and effective moduli. However, in sedimentary rocks, wave velocities and attenuation rates are dominated by pore-fluid effects, such as poroelasticity, squirt, and mesoscopic wave-induced fluid flows. Physically, such effects are significantly different from viscoelasticity, and the pore-fluid and VE phenomena are difficult to compare quantitatively without a common theoretical framework. We develop such a unified macroscopic framework that we call the general linear solid (GLS). The GLS is based on Lagrangian continuum mechanics, and it can be summarized as multiphase poroelasticity extended by solid and fluid viscosities. The formulation is carried out strictly in terms of continuum mechanics, measurable physical properties, and boundary conditions, from which the observable wave velocities and attenuation are predicted. Explicit differential equations are derived in matrix form, from which a variety of numerical modeling schemes can be obtained. A rigorous correspondence principle is formulated, in which viscosity effects contribute to complex-valued VE moduli, and Darcy friction lead to a complex-valued density matrix. Within the GLS framework, the viscoelasticity represents an end member characterized by zero Darcy-type friction, whereas the poroelasticity is an end member with zero solid viscosity. Transitions between these end members and their extensions yield macroscopic models of viscoporoelasticity, poroelasticity with multiple saturating fluids and double porosity, and poroelasticity with squirt flows. The approach is illustrated on models of layered poroelastic and viscoporoelastic media. Applications of the GLS framework are continued in part 2 of this study.