* Pan-European STOXX 600 index down 0.3 pct
* Banks down after Draghi remarks, real estate up
* Unilever inches up after strategy review
* Shares off lows as commodity stocks recover (Adds details, updates prices)
By Danilo Masoni
MILAN, April 6 (Reuters) - European shares fell on Thursday after minutes of the Federal Reserve's last meeting raised worries over high stock market valuations, with banks hit by cooling ECB rate hike expectations.
The STOXX 600 index fell 0.3 percent while Britain's FTSE and Germany's DAX both fell 0.4 percent.
European Central Bank chief Mario Draghi said the bank would stick to its policy plan including bond buying and record-low rates for some time to come as it was not yet convinced the euro zone economy was back to rude health.
His remarks, which suggested the ECB will not change its policy message this month, hit banking stocks, which benefit from prospects that higher rates could boost their margins, but lifted the real estate index to a six-month high.
"Banks have been suffering from negative newsflow on the rates front with the central bank rapidly becoming more prudent after inflation data resoundingly missed expectations," said Giuseppe Sersale, fund manager at Anthilia in Milan. "On the other hand, real estate is recovering from worries that rates could start a phase of strong increases."
Among real estate stocks, France's Unibail Rodamco provided the biggest boost to the sectoral index with a rise of 1.1 percent. Among euro zone banks, the top two fallers were Italy's Banco BPM and Germany's Commerzbank, both down 1.6 percent. Deutsche Bank fell 1 percent on the last day to subscribe to its 8 billion euros cash call.
Anthilia's Sersale added that broader risk-off mood was also due to profit-taking after a strong gain in March and a reversal at Wall Street following the release of the Fed minutes overnight.
Unilever edged up 0.2 percent after the consumer goods giant promised a multi-billion pound programme of shareholder rewards after a corporate rethink sparked by a takeover approach from Kraft Heinz.
Eslewhere broker upgrades drove price action.
Pearson was the top loser on the STOXX, down 8.7 percent, as the British education firm traded ex-dividend and was further weighed down by a downgrade from Exane to underperform.
AstraZeneca fell 1.3 percent after a UBS downgrade to neutral, while German airport operator Fraport rose 2.9 percent with traders linking the rise to a upgrade to "buy" from Societe Generale.
European shares come off lows as commodity stocks recovered after crude oil prices turned higher after being hit in the previous session record high U.S. crude inventories. (Reporting by Danilo Masoni; Editing by Alison Williams)