There are a lack of proper adhesives that meet the wood industry requirements of being environmentally friendly, low cost, and easy to use; thus, the application of thermoplastic polymers, especially films, is promising. This work expands our knowledge about the possibility of using thermoplastic films for the production of environmentally friendly plywood. The effects of the adhesives type and wood species on the properties of plastic film bonded plywood were studied. Sliced veneers of two hardwoods (birch and beech) and one softwood (spruce) were used in the experiments. Three types of thermoplastic films—low-density polyethylene (LDPE), co-polyamide (CoPA), and co-polyester (CoPE)—were used as an adhesive for bonding plywood samples. Melamine–urea–formaldehyde (MUF) resin was used as a reference. The influence of the type of adhesive and wood species as well as their interaction on the properties of plywood was significant. The lowest bonding strength demonstrated plywood samples bonded by LDPE, and the highest bonding strength in the samples was shown in those bonded by CoPA. A significant difference was found between softwoods and hardwoods in terms of their influence on the physical and mechanical properties of plywood samples. From the obtained data, it follows that softwoods provide much lower values of bending strength (MOR), modulus of elasticity (MOE), and bonding strength than hardwoods. The obtained bonding strength values of plastic-bonded plywood panels ranged from 1.18 to 2.51 MPa and met the European standard EN 314-2 for Class 1 (dry conditions) plywood.