Depends a bit on your desired "look", budget and method of working.
Softwood will be most economical, if you were going for put and out cheapness and have a Planer Thicknesser and powered sander then sawn whitewood carcassing would be fine. For a Hand-tool approach unsorted redwood is much nicer to work and achieve a good finish by hand. Softwoods have a distinctive knotty appearance and pronounced differences in early wood and latewood which are not to all tastes through.
In traditional hardwoods, ash would give a cleaner look at a reasonable price, and beech would give a very clean look at a slight increase in expense. Oak and elm both give a very traditional aesthetic but further increase expense (and with oak, preclude using iron or steel fittings), for a dark look without staining black walnut is the only choice in this catagory but is as expensive as many fine exotic timbers.
Finally exotics give a broad colour pallet, but also the greatest scope for expense, Keruing is cheap, strong and tough with a lovely light brown colour, but hard to achieve a fine finish and tough on cutting edges, Meranti comes in a light and dark shade of red-brown with the darker being rather stronger (and more expensive), it also has lovely working characteristics, Mahogany is at the expensive end of the market, (still cheaper than oak though!) But as it's now from plantation sources and used mainly in fine joinery/furniture it's not as commonly sold in the large sectional sizes you might want. Darker woods start to run into issues with expense or working characteristics (often both) but look wonderful. Good prices can be had sometimes on very unusual timbers like purpleheart, but whether you'd elect to use them is rather another matter.
What your OP causes me to envisage is a simple, rectilinear bed made of 4"×4" or 6"×6" legs and 2"×8" stretchers with 1"×3" boards forming a slatted headboard and footboard, made from elm or ash and finished with a mildly coloured varnish which imparts an aged yellowish-brown hue.