Is it possible to "naturally" reach the first three jhanas Is it possible, having been taught meditation by a zen master, to naturally enter the 1st three jhanas. By that, I mean through mental effort which is not really meditation as we normally think of it (seated or walking, all effort being applied to that).

I ask because the 1st three jhanas seem, potentially, much more mundane than the rest, worldly. e.g. I feel pretty strong sensations of pleasure most of the time, as long as I'm not there and then planning to do something.

If that second paragraph seems super delusional, then how close is dhyana to a pleasure in which you can't identify why you feel good?
All dogs are animals but all animals are not dogs. Similarly, each of the 1st three rupa jhanas has pleasure as a factor however all feelings of pleasure are not jhana. Each jhana is characteristized by ekaggatā (one-pointedness; stillness; unmovingness) rather than by feelings of pleasure.
That said, jhana is not reached by 'effort'. It is reached by 'letting go', as quoted below:

And what is the faculty of concentration? There is the case where a monk, a noble disciple, making it his object to let go (vossagga),
attains concentration, attains singleness of mind.
SN 48.10

Therefore, when you say: &quot;I feel pretty strong sensations of pleasure most of the time, as long as I'm not there and not planning to do something&quot;, this is the 'natural' path to jhana the Lord Buddha described in SN 48.10.
This is similar to the Zen masters, for example:

The Supreme Way is not difficult If only you do not pick and choose. Neither love nor hate, And you will clearly understand. Be off by a
hair, And you are as far from it as heaven from earth. If you want the
Way to appear, Be neither for nor against. For and against opposing
each other This is the mind's disease. Without recognizing the
mysterious principle It is useless to practice quietude. The Way is
perfect like great space, Without lack, without excess. Because of
grasping and rejecting, You cannot attain it.
Hsin Hsin Ming

Buddhadasa, Thai teacher, who fancied Zen, spoke a lot about natural concentration:

As for samadhi, an empty mind is the supreme samadhi, the supremely focused firmness of mind. The straining and striving sort of samadhi
isn't the real thing and the samadhi which aims at anything other than
non-clinging to the five khandas is micchasamadhi (wrong or perverted
samadhi). You should be aware that there is both micchasamadhi and
sammasamadhi (right or correct samadhi). Only the mind that is empty
of grasping at and clinging to 'I' and 'mine' can have the true and
perfect stability of sammasamadhi. One who has an empty mind has
correct samadhi.  Heartwood from the Bo Tree

Also refer, here: http://www.accesstoinsight.org/tipitaka/an/an11/an11.002.than.html
http://www.buddhanet.net/budasa10.htm