Pete, construction adhesives will not provide the bond strength which can be achieved in PVA wood glue (titebond types) or moisture cured urethane (gorrila glue types) provided your joints are well fitted and are properly clamped while the glue is being set. Construction adhesive will most likely provide a superior bond over the other 2 if you have gaps in your joints or cannot provide adequate clamp pressure.
So, here's what I would recommend:
For tight fitting joints with adquate clamp pressure:
1- If you outside furniture will only be exposed to rain intermintently, I'd say go with the Titebond III. It's designated as a 1 part Type 1 PVA, and I'm probably going to get into trouble here, but I would say that it will have difficulty meeting that spec.
2- If your furniture will be totally exposed to rain go with the moisure cured urethane (gorrilla glue types). For this application I like to spritz both wood surfaces with water then apply the glue and assemble, something like an old windex bottle will work nicely. Keep in mind that this system relies on foaming action of the glue to drive the adhesive into the wood pores, so you need very tight fitting joints and strong clamping pressure. Glueline performance will be degraded if the glueline is more than several mils thick. Properly done, this system will most likely provide 100% wood failure in a destructive test, and will easily meet a Type 1 spec.
If you feel you can't get good fitting joints or adequate clamp pressure on your joints you're probably best off with one of the better construction adhesives.
Hope this helps, let us know which way you go.